COLLUSION VIA RESALE By Rodney J. Garratt, Thomas Troger,¨ and Charles Z. Zheng1 The English auction is susceptible to tacit collusion when post-auction inter- bidder resale is allowed. We show this by constructing equilibria where, with positive probability, one bidder wins the auction without any competition and divides the spoils by optimally reselling the good to the other bidders. These equilibria interim Pareto dominate (among bidders) the standard value-bidding equilibrium, without requiring the bidders to make any commitment on bidding behavior or post-bidding spoil-division. Keywords: English auction, second-price auction, collusion, sunspots, resale. 1. INTRODUCTION In private-value English auctions that ban resale, it is a dominant strategy for each partic- ipant to bid up to her use value. With resale allowed, value-bidding remains an equilibrium outcome, but there is no dominant strategy. Resale opens the possibility that some bidders will optimally drop out at a price below their use values. They prefer to let a competitor win and buy from her in the resale market. The existence of non-value-bidding equilibria is important because the celebrated advantages of the English auction, in particular efficiency, are based on value-bidding, and because resale is possible in most applications. In this paper we construct a family of non-value-bidding equilibria for an English auc- tion that allows inter-bidder resale. Such equilibria exist in any independent private value environment (symmetric or asymmetric) for any number of bidders (Proposition1). Each equilibrium in this family is identified by the choice of a designated bidder and a threshold type, below which all bidders, except the designated bidder, bid zero. All bidders with types 1We thank Subir Bose, Paul Heidhues, George Mailath, Leslie Marx, Tymofiy Mylovanov, Greg Pavlov, Larry Samuelson and three anonymous referees for helpful comments. We are particularly grateful to Dan Levin for suggesting that we investigate the collusive properties of our equilibrium construction. 1 above the threshold bid up to their values. In cases where the designated bidder wins the initial auction and has a sufficiently low type, she will offer the item for resale instead of consuming it. Because the determination of the designated bidder does not depend upon her type and the resale market retains information asymmetry, the final outcome may be inefficient. Since a designated bidder may win the initial auction at a low price, such equilibria provide an opportunity for a form of tacit collusion among the bidders. By using a publicly observed randomizing device (or sunspot) to choose the designated bidder, the surplus can be distributed in a way that makes every bidder of every type better-off than under the value- bidding equilibrium; i.e., the value-bidding equilibrium is interim (bidder-)Pareto dominated (Proposition2). 2 The recommendation made by the sunspot device is not binding. Once the sunspot picks a designated bidder, it is in the interest of each bidder to bid accordingly in the initial auction based on the expectation that others will follow their assigned roles. Previous models of collusion in second-price and English auctions (e.g., Graham and Marshall, 1987, Mailath and Zemsky, 1991, Marshall and Marx, 2007) rely on pre-auction communication, in which every colluding bidder reports her type to the bidding ring. By communicating, the colluding bidders determine side payments and designate a single bidder to participate in the auction and win at a low price. These papers specify mechanisms that achieve efficient collusion as an equilibrium. However, the proposed use of pre-auction communication is problematic because it is usually illegal and participating bidders risk being detected. Moreover, the proposed collusive schemes require the non-designated bidders in the bidding ring to bid below their values in the actual auction. Without resale, such bidding strategies are weakly dominated, and bidders may not be willing to play them. To make the expectation of a dominated strategies credible, the colluding bidders might require a commitment device.3 2Readers who are familiar with U.S. litigation history might draw some parallels between our proposed use of a sunspots variable and the famous phases-of-the-moon bidding ring that was operated by electrical equipment suppliers in the 1950s. The phases-of-the-moon scheme earned its designation because it involved an explicit two-week rotation to determine the low bidder. See Smith (1961). 3Sustaining collusion in first-price auctions is more difficult than in second-price auctions because non- designated bidders have a strict incentive to overbid the designated bidder whose bid in the main auction is 2 By introducing the possibility of resale after an English or second-price auction, our paper rationalizes collusion without pre-auction communication or dominated strategies. Instead of pre-auction communication of private information, a publicly observable sunspot selects a designated bidder in a manner commonly known to the colluding bidders, and the final owner of the good is decided through a resale mechanism. Before the auction, no one commits to what she will do in the auction or at resale. During the auction, colluding bidders do not bid up to their values, however we prove that such strategies are not weakly dominated given the option for resale (Appendix B). The winner of the auction chooses a resale mechanism that is optimal for her given the posterior beliefs after the auction, and only after the initial auction has ended can she commit to the rules of her resale mechanism. McAfee and McMillan (1992, p. 587) noted that in practice a bidding ring's own \knock- out auction" often happens after rather than before the legitimate auction. This practice is well represented in our equilibria. Our Pareto-improving equilibria are, in contrast to the previously proposed collusive schemes, not ex-post efficient.4 This is consistent with an impossibility result in Lopomo, Marshall, and Marx (2005), which shows that inefficiency is a quite general feature of Pareto- improving equilibria in English auctions without pre-auction communication. From the view- point of antitrust authorities, inefficient collusive equilibria are important precisely because of the distortion; efficient collusion involves merely a pure transfer from the seller to the bidders. Blume and Heidhues (2004) completely characterize the Bayesian Nash equilibria for the second-price auction with three or more bidders with a common type space. These equilibria are valid for English auctions and have the same bidding structure as our equilibria. However, because there is no resale market, the Blume-Heidhues equilibria are in dominated strategies. Moreover, in some environments none of these equilibria Pareto dominates the value-bidding equilibrium, even if we give everyone a chance to be the designated bidder through sunspot coordination. This is because high-value bidders strictly prefer the value- bidding equilibrium; we show that for all symmetric environments with strictly concave value below their value; see McAfee and McMillan (1992) and Marshall and Marx (2007). 4The payoff gains in our equilibria relative to value-bidding can still be substantial (Table I, Section 5. 3 distributions (Proposition3). But with resale, there always exists an equilibrium that makes every type of every bidder strictly better off than the value-bidding equilibrium. In contrast to much of the earlier literature on auctions with resale, our model allows for any number of asymmetric bidders. With multiple bidders at the resale stage, the optimal resale mechanism typically is no longer a take-it-or-leave offer, as often assumed, but is an optimal auction as derived by Myerson (1981). A technical novelty of our paper is that we avoid complicated explicit computations of bidders' resale payoffs. We identify several structural properties of the resale continuation game that facilitate our perfect Bayesian equilibria for the entire auction-with-resale game. These structural properties do not appear to be specific to the Myerson optimal auction, suggesting that our qualitative results extend to other forms of the resale market.5 Our result that the value-bidding equilibrium is interim (bidder-)Pareto dominated is based on two regularity properties of resale that are satisfied in our equilibria. The properties ensure that, even when the threshold is arbitrarily close to zero, a non-vanishing fraction of the bidder-types below the threshold still engage in actual resale trade. The proof begins with the observation that our equilibria interim Pareto dominate the value-bidding equilibrium if the prior type distributions are uniform. Then we extend this dominance relation to arbitrary type distributions provided that the thresholds in our equilibria are sufficiently small. Sufficiently small thresholds allow us to approximate the expected payoffs by the ones in the uniform-distribution case. The aforementioned regularity properties imply that a bidder's gain from trade at resale outweighs the error of the approximation. The threshold-bidding strategies in our equilibria are built upon Garratt and Tr¨oger(2006) for English and second-price auctions. However, there are nontrivial differences. In Garratt and Tr¨oger,the bidder who will become the reseller has no private information, and the other bidders have identical prior value distributions. In contrast, in this paper every bidder has private information and bidders' value-distributions can differ. Our extension of the equilibrium construction to the case of asymmetric bidders is made possible by conditioning 5For instance, the period-2 seller may be restricted to use a second-price or English auction with an optimal reserve price, or the period-2 seller may use an English auction with the right to reject all bids (Haile, 2003), or, in two-bidder environments, a random draw may specify which bidder has the right to propose a resale price (Calzolari and Pavan, 2006). 4 the designated bidder's bid on the identities of the bidders who stay in the auction.